The area is unbiased and the perimeter is not

A length measured from noisy points is too long established the previous rung’s result: the distance between two coordinates each carrying isotropic error of standard deviation σ has expectation

$$\mathbb{E}[\text{measured}] - d \approx \frac{\sigma^2}{d},$$

always positive, always in the same direction, and invisible in any error budget that stops at first order. The transverse component is the whole of it — a perturbation along the baseline cancels between the two ends and a perturbation across it does not.

The obvious next question is what that does to an area, since a boundary has both. The answer is that it does nothing at all, and the reason is worth the essay.

Why the area does not move #

The shoelace formula for the area of a closed polygon is

$$A = \tfrac12\left|\sum_i \left(x_i y_{i+1} - x_{i+1} y_i\right)\right|,$$

and every term in that sum is a product of coordinates belonging to two different vertices. If the errors on the vertices are independent, the expectation of each product is the product of the expectations, so every term comes out at its true value and the sum does too.

The shoelace area is exactly unbiased. Not approximately, not to second order, not for small σ — exactly, for any noise distribution whatsoever with the right mean, as long as adjacent vertices are independent.

The contrast with the perimeter is entirely about the shape of the functional. A length is $\sqrt{\Delta x^2 + \Delta y^2}$, which is a concave function of the displacement, and Jensen’s inequality does the rest: the expectation of a concave function is below the function of the expectation, which for a distance means the average measured length exceeds the true one. Area is a polynomial, and a bilinear one, with no square root anywhere.

The area’s spread, which is not small #

Unbiased is not the same as accurate, and the area’s variance has a closed form worth having.

Differentiating the shoelace with respect to one vertex gives half the vector joining its two neighbours, so under independent isotropic noise of standard deviation σ,

$$\operatorname{Var}(A) = \frac{\sigma^2}{4}\sum_i \left|P_{i+1} - P_{i-1}\right|^2.$$

That is the formula the simulation is checked against, and it agrees to a twentieth of a per cent at every vertex count tried. It also says something the simulation would take a while to notice: the area’s precision depends on the spacing between alternate vertices, so on a boundary whose vertices are close together, each one hardly matters.

Which way densifying moves each of them #

Densifying a boundary — putting more vertices along the same line — is the standard response to almost every geometric complaint this collection has raised. It makes a stored segment follow the geodesic it claims, it makes an intersection land in the right place, it makes a projected page area converge.

Here it does two things at once, in opposite directions.

Both rates follow from the two formulas and neither is a surprise once they are written down. The area’s variance sums $|P_{i+1} - P_{i-1}|^2$, and doubling the vertex count halves each of those vectors while doubling their number, so the sum halves and the standard deviation falls as $1/\sqrt{n}$. The perimeter’s bias sums $\sigma^2/d$ over the legs, and doubling the count halves $d$ while doubling the number of terms, so the total quadruples.

A quarter of a metre of spurious perimeter on a 628-metre boundary is 415 parts per million, from a survey whose vertices are good to five centimetres. No specification is written to catch that, and the operation that produced it was an improvement.

What was computed, and how #

A regular polygon, because its area, perimeter and area-variance all have closed forms and nothing depends on a dataset. Independent isotropic Gaussian noise on each vertex, drawn antithetically — each perturbation used with both signs — so the odd-order terms cancel exactly rather than statistically. That is the same device the previous rung needed and for the same reason: the quantity being measured is second order and it lives inside a first-order spread.

One thing had to be repaired to make the variance measurable at all, and it is worth recording because it looked like a formula error for an hour. The generator this file had used since the sixth rung is a linear congruential one whose multiplier is 1,103,515,245; multiplied by a 32-bit state that reaches 4.7 × 10¹⁸, which is past the 2⁵³ a double carries exactly, so the low bits of every draw are rounding rather than state. It is good enough for a mean and not good enough for a variance: the measured standard deviation of the area came out seven per cent below the closed form and stayed there at every sample size, which is the signature of a generator rather than of an integrand. Doing the multiplication in 32-bit integers removes the disagreement entirely.

The assertions require three things and each could fail on its own: the area’s bias must be inside three standard errors of zero, the perimeter’s must be outside eight, and the area’s measured spread must match the closed form to five per cent.

What a compactness score inherits #

The place this matters most is not an area or a perimeter but a ratio of them. Shape compactness is scored by the isoperimetric quotient

$$C = \frac{4\pi A}{P^2},$$

which is one for a disc and less for anything else, and which is used to score administrative districts, to detect gerrymandered boundaries, and to describe the shape of catchments and lakes and cities.

Its numerator is unbiased and its denominator is biased high, so the score is biased low, always, and by twice the perimeter’s relative bias.

At 256 vertices and 20 centimetres of noise the score is 1.3 per cent low, which is comfortably enough to reorder a list of districts. And the direction of the effect is the awkward part: a boundary digitised more finely scores as less compact, from the same ground, with the same instrument, with nothing wrong anywhere. A comparison between two districts whose boundaries were captured at different resolutions is partly a comparison of the capture.

The most compact shape depends on the paper prices the other half of the same score — that both $A$ and $P$ are page quantities and neither survives a projection. The two failures are independent, they have different signs on different projections, and a published compactness figure carries both.

Which error model spoils which statistic #

None of this makes the area a safe statistic. It makes the area safe against one error model, and the models are not interchangeable.

A shared shift spoils neither, and it is the control: both statistics see only differences between vertices, so a common translation cancels identically. That is worth stating because it is the error a datum problem produces, and it is the largest error most datasets have. The difference of two coordinates is the general form of the same observation.

A random scale factor spoils the area and not the perimeter, and it reverses the whole rung. An area is quadratic in a scale, so a scale factor with mean one and relative standard deviation $s$ biases the area high by exactly $s^2$ — measured 4.00 × 10⁻⁶ at $s$ = 0.002, which is $s^2$ to three figures. A perimeter is linear in a scale and comes out unbiased to fourteen decimal places.

So the two statistics have opposite sensitivities to the two commonest error structures in survey work, and neither is the safe one. Which statistic to trust is decided by which error is present, and that is a question about the survey rather than about the geometry.

The one place the independence is guaranteed to fail #

There is a systematic reason to expect the independence assumption to break, and it is not sloppiness in the field. It is the adjustment.

A coordinate is the output of a solve: the positions in a modern survey come out of a least-squares adjustment of a network of observations, and an adjustment produces a full covariance matrix. Neighbouring stations in a network share observations, share control, and share the constraints that fixed the datum, so their errors are correlated — strongly, and by construction. What another common point buys measures the same structure from the transformation side.

The correlation is mostly of the harmless kind. A network’s dominant error modes over a small area are a shift and a rotation, both of which cancel exactly in an area and in a perimeter, which is why cadastral areas are as reliable as they are. But an adjustment also produces a scale mode, and a scale mode is the one that biases the area — so the safest-looking statistic on a boundary is the one exposed to the error a network is least good at controlling, which is the network’s own scale.

That is not a reason to distrust registered areas. It is a reason not to describe the area as the unbiased statistic without saying what it is unbiased against, which is the trap this rung exists to name.

What a specification would have to say #

A boundary specification states a positional tolerance: every vertex within so many centimetres of truth. That single number is what a client buys and it is what a contractor is judged against, and it is silent about everything in this essay.

It does not bound the perimeter’s error, because the perimeter’s bias depends on the leg length as well as on σ, and the leg length is a decision the contractor makes. Two surveys meeting the same five-centimetre tolerance on the same 628-metre boundary differ by a quarter of a metre in reported length if one captured 256 vertices and the other four.

It does not bound the area’s error either, though for the opposite reason: the area has no bias to bound, and its spread is set by the closed form above, which the specification never mentions. A client wanting an area to a stated precision needs a statement about $\sum|P_{i+1}-P_{i-1}|^2$, and no specification anywhere is written that way.

And it cannot bound a compactness score at all, because the score’s error is dominated by a bias whose size depends on a capture decision.

The repair, in each case, is to specify the derived quantity rather than the vertices, which is what a specification is actually about. That is the same move the tolerance decides the model makes for a survey’s method: state the number the job needs, and let the method follow from it.

Where the model stops #

Independence between adjacent vertices is doing all the work. Real boundary vertices are usually observed from the same instrument set-up, referred to the same control, and captured by the same operator in the same pass, so their errors are correlated — and any correlation moves the area’s bias off zero. The direction depends on the correlation’s structure, and the two structures measured here bracket it: a pure translation leaves the area alone and a pure scale biases it high.

A polygon on the plane. Every formula here is planar, so the area is a page area and everything computing an area needs a surface says about the difference between that and a ground area still applies, on top of all of this.

And nothing here is about a blunder. These are the effects of well-behaved noise on a well-observed shape. The blunder the network cannot see is a different failure with different arithmetic, and a boundary carrying one has a much larger problem than a quarter of a metre of perimeter.

The generalisation #

The rule is short and it generalises past geometry: a nonlinear functional of noisy inputs is biased, and the sign of the bias is the sign of its curvature.

A concave functional — a distance, a square root, a minimum — comes out too small in its argument and therefore, for a distance, too large in its answer. A convex one — a square, an area under a scale, a variance — comes out too large. A multilinear one — a shoelace, a determinant, a covariance between independent quantities — comes out exactly right, because linearity in each argument separately is all the independence needs.

That is why the two halves of one boundary behave so differently, and it is a much more useful classification than “areas are more robust than lengths”, which is the folklore version and is true only for the error model people happen to have in mind. Check the curvature, and check which arguments the noise is independent between. Everything in this essay follows from those two questions and neither of them mentions maps.

Who found it, and when #

The bias of a measured distance is old and belongs to the surveying literature; the previous rung gives its history. The unbiasedness of the shoelace under independent vertex noise is a two-line consequence of bilinearity that appears, without fanfare, in the geographic information science literature on positional error propagation — Chrisman and Yandell gave the area variance formula in 1988, and it is the closed form checked here.

What is not in either place is the pair. The area result is quoted as reassurance — areas are robust to digitising error — and the length result is quoted as a caution, and the two live in different papers about different operations. Putting them beside each other says that the same boundary yields one trustworthy number and one untrustworthy one, and that the standard remedy for geometric complaints makes the split worse rather than better.

The compactness consequence appears to be unremarked. Scores are published to three decimal places, compared across jurisdictions whose boundary data have different capture resolutions, and used in litigation.

A compactness score that does not inherit the bias #

The compactness consequence is the one with a use attached, so it is worth saying what the remedy is rather than leaving the caution unresolved.

The trouble is the perimeter and only the perimeter. The familiar scores put it in the numerator or the denominator — the isoperimetric ratio $4\pi A/P^2$ is the standard one — so they inherit a quantity that grows with vertex noise and grows further with densification. Two jurisdictions whose boundaries were captured at different resolutions get different scores for the same shape, and the direction of the difference is knowable in advance: the finer capture scores as less compact, always.

Scores built from areas alone do not have the problem. The ratio of a region’s area to the area of its convex hull, or to the area of the smallest enclosing circle, or to the area of the largest inscribed circle, are all compactness measures in the ordinary sense — each is one for a disc and falls towards zero for a straggling shape — and each is a ratio of two areas, both of which are unbiased under independent vertex noise for the same bilinearity reason the essay gives.

They are not free of noise: a hull or an enclosing circle depends on extreme vertices, which are the ones noise moves most, so the spread is larger than the area’s own. But the error is a spread rather than a drift, and a spread is something that averages away over many regions while a bias does not. That is the whole of the distinction this rung is about, applied to the statistic that has most often been built the other way.

And the practical test is available to anybody comparing two jurisdictions. Densify both boundaries to a common vertex spacing and recompute. A perimeter-based score moves; an area-based one barely does; and if the ranking of two regions changes under that operation, the ranking was a measurement of capture resolution.

Where the ladder goes next #

Eight rungs have taken a coordinate’s width through a projection, an average, a difference, a distance and now an enclosed area. What every one of them has assumed is that the noise is on the positions. A boundary has another kind of uncertainty that no covariance describes: whether a vertex belongs to the boundary at all, which is a question about the feature rather than about the measurement, and which the anchor on generalisation has been answering from the other side.